The Banach-Tarski Paradox

<h2>A Brief History</h2> <p>The Banach-Tarski Paradox was first published in 1924 by Banach and Tarski in their groundbreaking paper,&nbsp;<a href="https://bibliotekanauki.pl/articles/1385774.pdf" rel="noopener ugc nofollow" target="_blank"><em>&ldquo;Sur la d&eacute;composition des ensembles de points en parties respectivement congruentes&rdquo;</em></a>&nbsp;(<em>On the decomposition of sets of points into congruent parts</em>). Their work was based on earlier discoveries by Austrian mathematician Felix Hausdorff, who had demonstrated a similar paradoxical decomposition for certain subsets of Euclidean space. Banach and Tarski extended Hausdorff&rsquo;s work to prove their astonishing result for the 3-dimensional sphere.</p> <h2>The Paradox</h2> <p>The Banach-Tarski Paradox is a consequence of the&nbsp;<a href="https://core.ac.uk/download/pdf/82763200.pdf" rel="noopener ugc nofollow" target="_blank"><em>Axiom of Choice</em></a>, an important and controversial principle in set theory. While the&nbsp;<strong><em>Axiom of Choice</em></strong>&nbsp;has many useful applications in mathematics, it also leads to some perplexing results, such as the Banach-Tarski Paradox.</p> <p><a href="https://www.cantorsparadise.com/the-banach-tarski-paradox-82bafb306e5d"><strong>Website</strong></a></p>